Compute the number of ordered triples $(x,y,z)$ of integers satisfying \[x^2+y^2+z^2=9.\] [i]Proposed by Nathan Xiong[/i]

Calculate the sum $$S=\sqrt{1+\frac{8\cdot 1^2-1}{1^2\cdot 3^2}}+\sqrt{1+\frac{8\cdot 2^2-1}{3^2\cdot 5^2}}+...+ \sqrt{1+\frac{8\cdot 1003^2-1}{2005^2\cdot 2007^2}}$$

Let $ABC$, $AC \not= BC$, be an acute triangle with orthocenter $H$ and incenter $I$. The lines $CH$ and $CI$ meet the circumcircle of $\bigtriangleup ABC$ at points $D$ and $L$, respectively. Prove that $\angle CIH = 90^{\circ}$ if and only if $\angle IDL = 90^{\circ}$

Let n numbers $x_1, x_2, \ldots, x_n$ be chosen in such a way that $1 \geq x_1 \geq x_2 \geq \cdots \geq x_n \geq 0$. Prove that \[(1 + x_1 + x_2 + \cdots + x_n)^\alpha \leq 1 + x_1^\alpha+ 2^{\alpha-1}x_2^\alpha+ \cdots+ n^{\alpha-1}x_n^\alpha\] if $0 \leq \alpha \leq 1$.

Fix positive integers $n$ and $k$ such that $2 \le k \le n$ and a set $M$ consisting of $n$ fruits. A [i]permutation[/i] is a sequence $x=(x_1,x_2,\ldots,x_n)$ such that $\{x_1,\ldots,x_n\}=M$. Ivan [i]prefers[/i] some (at least one) of these permutations. He realized that for every preferred permutation $x$, there exist $k$ indices $i_1 < i_2 < \ldots < i_k$ with the following property: for every $1 \le j < k$, if he swaps $x_{i_j}$ and $x_{i_{j+1}}$, he obtains another preferred permutation. \\ Prove that he prefers at least $k!$ permutations.

Let $\alpha_1, \alpha_2, \dots, \alpha_n$, and $\beta_1, \beta_2, \ldots, \beta_n$, where $n \geq 4$, be 2 sets of real numbers such that \[\sum_{i=1}^{n} \alpha_i^2 < 1 \qquad \text{and} \qquad \sum_{i=1}^{n} \beta_i^2 < 1.\] Define \begin{align*} A^2 &= 1 - \sum_{i=1}^{n} \alpha_i^2,\\ B^2 &= 1 - \sum_{i=1}^{n} \beta_i^2,\\ W &= \frac{1}{2} (1 - \sum_{i=1}^{n} \alpha_i \beta_i)^2. \end{align*} Find all real numbers $\lambda$ such that the polynomial \[x^n + \lambda (x^{n-1} + \cdots + x^3 + Wx^2 + ABx + 1) = 0,\] only has real roots.

In the diagram below, fill the $12$ circles with numbers from the following bank so that each number is used once. Two circles connected by a single line must contain relatively prime numbers. Two circles connected by a double line must contain numbers that are not relatively prime. $$\text{Bank: } 20, 21, 22, 23, 24, 25, 27, 28, 30 ,32, 33 ,35$$ [asy] real HRT3 = sqrt(3) / 2; void drawCircle(real x, real y, real r) { path p = circle((x,y), r); draw(p); fill(p, white); } void drawCell(int gx, int gy) { real x = 0.5 * gx; real y = HRT3 * gy; drawCircle(x, y, 0.35); } void drawEdge(int gx1, int gy1, int gx2, int gy2, bool doubled) { real x1 = 0.5 * gx1; real y1 = HRT3 * gy1; real x2 = 0.5 * gx2; real y2 = HRT3 * gy2; if (doubled) { real dx = x2 - x1; real dy = y2 - y1; real ox = -0.035 * dy / sqrt(dx * dx + dy * dy); real oy = 0.035 * dx / sqrt(dx * dx + dy * dy); draw((x1+ox,y1+oy)--(x2+ox,y2+oy)); draw((x1-ox,y1-oy)--(x2-ox,y2-oy)); } else { draw((x1,y1)--(x2,y2)); } } drawEdge(2, 0, 4, 0, true); drawEdge(2, 0, 1, 1, true); drawEdge(2, 0, 3, 1, true); drawEdge(4, 0, 3, 1, false); drawEdge(4, 0, 5, 1, false); drawEdge(1, 1, 0, 2, false); drawEdge(1, 1, 2, 2, false); drawEdge(1, 1, 3, 1, false); drawEdge(3, 1, 2, 2, true); drawEdge(3, 1, 4, 2, true); drawEdge(3, 1, 5, 1, false); drawEdge(5, 1, 4, 2, true); drawEdge(5, 1, 6, 2, false); drawEdge(0, 2, 1, 3, false); drawEdge(0, 2, 2, 2, false); drawEdge(2, 2, 1, 3, false); drawEdge(2, 2, 3, 3, true); drawEdge(2, 2, 4, 2, false); drawEdge(4, 2, 3, 3, false); drawEdge(4, 2, 5, 3, false); drawEdge(4, 2, 6, 2, false); drawEdge(6, 2, 5, 3, true); drawEdge(1, 3, 3, 3, true); drawEdge(3, 3, 5, 3, false); drawCell(2, 0); drawCell(4, 0); drawCell(1, 1); drawCell(3, 1); drawCell(5, 1); drawCell(0, 2); drawCell(2, 2); drawCell(4, 2); drawCell(6, 2); drawCell(1, 3); drawCell(3, 3); drawCell(5, 3); [/asy]

A right cylinder is given with a height of $20$ and a circular base of radius $5$. A vertical planar cut is made into this base of radius $5$. A vertical planar cut, perpendicular to the base, is made into this cylinder, splitting the cylinder into two pieces. Suppose the area the cut leaves behind on one of the pieces is $100\sqrt2$. Then the volume of the larger piece can be written as $a + b\pi$, where $a, b$ are positive integers. Find $a + b$.

The vertices A,B,C of an acute-angled triangle ABC lie on the sides B1C1, C1A1, A1B1 respectively of a triangle A1B1C1 similar to the triangle ABC (∠A = ∠A1, etc.). Prove that the orthocenters of triangles ABC and A1B1C1 are equidistant from the circumcenter of △ABC.

$ABC$ is a triangle with orthocenter $H$. The median from $A$ meets the circumcircle again at $A_1$, and $A_2$ is the reflection of $A_1$ in the midpoint of $BC$. The points$ B_2$ and $C_2$ are defined similarly. Show that $H$, $A_2$, $B_2$ and $C_2$ lie on a circle. [img]https://cdn.artofproblemsolving.com/attachments/f/1/192d14a0a7a9aa9ac7b38763e6ea6a4a95be55.png[/img]

There are integers $a$, $b$, and $c$, each greater than 1, such that $$\sqrt[a]{N \sqrt[b]{N \sqrt[c]{N}}} = \sqrt[36]{N^{25}}$$ for all $N > 1$. What is $b$? $\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6$

Initially, $10$ ones are written on a blackboard. Grisha and Gleb are playing game, by taking turns; Grisha goes first. On one move Grisha squares some $5$ numbers on the board. On his move, Gleb picks a few (perhaps none) numbers on the board and increases each of them by $1$. If in $10,000$ moves on the board a number divisible by $2023$ appears, Gleb wins, otherwise Grisha wins. Which of the players has a winning strategy?

Write the equation of the line tangent to the graph of the function $y = x^4-x^2 + x$ to at least at two points.

Let us define a $\textit{triangle equivalence}$ a group of numbers that can be arranged as shown $a+b=c$ $d+e+f=g+h$ $i+j+k+l=m+n+o$ and so on... Where at the $j$-th row, the left hand side has $j+1$ terms and the right hand side has $j$ terms. Now, we are given the first $N^2$ positive integers, where $N$ is a positive integer. Suppose we eliminate any one number that has the same parity with $N$. Prove that the remaining $N^2-1$ numbers can be formed into a $\textit{triangle equivalence}$. For example, if $10$ is eliminated from the first $16$ numbers, the remaining numbers can be arranged into a $\textit{triangle equivalence}$ as shown. $1+3=4$ $2+5+8=6+9$ $7+11+12+14=13+15+16$

A point is chosen at random within the square in the coordinate plane whose vertices are $(0, 0),$ $(2020, 0),$ $(2020, 2020),$ and $(0, 2020)$. The probability that the point is within $d$ units of a lattice point is $\tfrac{1}{2}$. (A point $(x, y)$ is a lattice point if $x$ and $y$ are both integers.) What is $d$ to the nearest tenth$?$ $\textbf{(A) } 0.3 \qquad \textbf{(B) } 0.4 \qquad \textbf{(C) } 0.5 \qquad \textbf{(D) } 0.6 \qquad \textbf{(E) } 0.7$

Let $a$, $b$ and $c$ be positive real numbers such that $a \geq b \geq c$. Prove the inequality: $\frac{a}{b}+\frac{b}{c}+\frac{c}{a} \leq \frac{b}{a}+\frac{c}{b}+\frac{a}{c}$

Determine all triples $(x, y, z)$ of nonnegative real numbers that verify the following system of equations: $$x^2 - y = (z -1)^2 $$ $$y^2 - z = (x -1)^2$$ $$z^2 - x = (y - 1)^2$$

Find the sum of all distinct prime factors of $2023^3 - 2000^3 - 23^3$. [i]Individual #5[/i]

The number $ \sqrt {2}$ is equal to: $ \textbf{(A)}\ \text{a rational fraction} \qquad\textbf{(B)}\ \text{a finite decimal} \qquad\textbf{(C)}\ 1.41421$ $ \textbf{(D)}\ \text{an infinite repeating decimal} \qquad\textbf{(E)}\ \text{an infinite non \minus{} repeating decimal}$

Let $p$ be a prime with $p>5$, and let $S=\{p-n^2 \vert n \in \mathbb{N}, {n}^{2}<p \}$. Prove that $S$ contains two elements $a$ and $b$ such that $a \vert b$ and $1<a<b$.

For three positive integers $a,b$ and $c,$ if $\text{gcd}(a,b,c)=1$ and $a^2+b^2+c^2=2(ab+bc+ca),$ prove that all of $a,b,c$ is perfect square.

Prove that every bijective function $ f: \mathbb{Z}\rightarrow\mathbb{Z}$ can be written in the way $ f\equal{}u\plus{}v$ where $ u,v: \mathbb{Z}\rightarrow\mathbb{Z}$ are bijective functions.

Four runners are preparing to begin a $1$-mile race from the same starting line. When the race starts, runners Alice, Bob, and Charlie all travel at constant speeds of $8$ mph, $4$ mph, and $2$ mph, respectively. The fourth runner, Dave, is initially half as slow as Charlie, but Dave has a superpower where he suddenly doubles his running speed every time a runner finishes the race. How many hours does it take for Dave to finish the race?

Prove that the affixes of three pairwise distinct complex numbers $ z_0,z_1,z_2 $ represent an isosceles triangle with right angle at $ z_0 $ if and only if $ \left( z_1-z_0 \right)^2 =-\left( z_2-z_0 \right)^2. $

There're $n$ students whose names are different from each other. Everyone has $n-1$ envelopes initially with the others' name and address written on them respectively. Everyone also has at least one greeting card with her name signed on it. Everyday precisely a student encloses a greeting card (which can be the one received before) with an envelope (the name on the card and the name on envelope cannot be the same) and post it to the appointed student by a same day delivery. Prove that when no one can post the greeting cards in this way any more: (i) Everyone still has at least one card; (ii) If there exist $k$ students $p_1, p_2, \cdots, p_k$ so that $p_i$ never post a card to $p_{i+1}$, where $i = 1,2, \cdots, k$ and $p_{k+1} = p_1$, then these $k$ students have prepared the same number of greeting cards initially.